C OMPOSITIO M ATHEMATICA
M ONTSERRAT T EIXIDOR I B IGAS
The divisor of curves with a vanishing theta-null
Compositio Mathematica, tome 66, no 1 (1988), p. 15-22
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Compositio Mathematica
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66: 15-22
(1988)
© Kluwer Academic Publishers, Dordrecht - Printed in the Netherlands
The divisor of
curves
with
a
vanishing theta-null
MONTSERRAT TEIXIDOR I BIGAS*
Department of Pure Mathematics, University of Liverpool, P.O.
Liverpool L69 3QX, UK
Received 6 March 1987;
§0.
accepted
Box 147,
19 November 1987
Introduction
by Jlg the moduli space of curves of genus g, by kg its natural
compactification by stable curves. We recall that Jlg-Jlg is the union of
[g/2] + 1 divisors Ao, A,1
0394[g/2]. A general point of Do is a curve obtained
two
in
a
curve of genus g - 1. A generic point Ai is
by identifying
points
obtained by identifying a point in a curve of genus i with a point in a curve
Denote
...
of genus g - i.
A theta-characteristic on a curve is a line bundle such that its square is the
dualizing sheaf. A non-singular curve has 22g theta-characteristics corresponding to the points of order two in its jacobian variety. A thetacharacteristic is said to be even or odd depending on whether the vector
space of its sections is even or odd. The parity of a theta-characteristic is
constant in any family ([M], [H]). Any non-singular curve has 2g-1 (2g + 1)
even and 2g-1(2g - 1) odd theta characteristics (see for instance [R, F] p. 4
and 176-177). The latter have necessarily a section. But a generic curve has
no theta-characteristics with 2 or more independent sections (as these line
bundles do not satisfy Gieseker-Petri Theorem). It follows that, for g
3,
the locus of curves which have an even theta characteristic with space of
sections of (projective) dimension at least one is a divisor in Jlg (see [Fa], [H],
[T]). Denote it by JI; and let JI; be its closure in g. The purpose of this
work is to prove that JI; is irreducible.
We note that any divisor in Jlg intersects 03941. The first step is to determine
the intersection of JI; and 03941. This is the union of 2 pieces (see (1.2)). One
of them is the locus of curves obtained by identifying a generic point in a
curve of u1g-1with a point in a generic elliptic curve. By induction hypothesis, this locus is irreducible. Moreover, the intersection of this piece with
any other component of JI; n 03941 contains certain degenerate hyperelliptic
*Supported by
C.S.I.C.
16
To finish the proof, we check then that the monodromy interchanges
dimensional effective theta-characteristics on these curves.
In the last paragraph of the this work, we compute the class of the divisor
in Ag.
Most notations will be standard. A curve will always be semistable, g will
denote its arithmetic genus. A (non necessarily invertible) torsion-free rankone sheaf on a (singular) curve which is the limit of theta-characteristics on
nearby curves will be called a limit theta characteristic. The dimension of a
(limit) theta-characteristic is the projective dimension of the space of
sections of the line bundle or torsion-free rank-one sheaf.
We shall assume that the reader is well acquainted with the theory of
Eisenbud and Harris of limit linear series on curves of compact type
([E, H]1) and that of admissible coverings of Harris and Mumford [H, M]).
curves.
the
even
1g
§1.
Détermination of the intersection of
u1g with the boundary components
of ug
Every component of the intersection of JI; with
has
codimension one in 0394i. This follows from the fact that u1g) has pure codimension one in ug and that Ãg is a normal variety. We shall also prove
that a generic curve in 0394i does not have a one dimensional (limit) thetacharacteristic. By a result of Harris ([H], Th. 1.10), every one dimensional
(limit) theta - characteristic extends to a one-dimensional (limit) thetacharacteristic on a locus of codimension at most one in every family of
curves containing the given curve. It follows that any curve with a (limit)
even theta-characteristic of dimension at least one lies in
u1g.
Hence, in order to determine the intersection of JI; and 0394i, one may
forget about those configurations of C which move in a locus of codimension 2 or more in Ai. Moreover, for loci of codimension one, it is only
necessary to determine if the generic curve in the locus lies in u1g.
(1.1).
REMARK.
a
Di
1 the intersection of u1g) with 0; consists of 4
(1.2). PROPOSITION. For i
The
curve
C
in
each piece is obtained from a curve Ç of genus
pieces.
generic
i and a curve Cg-i of genus g - i by identifying a point of each to a single point,
say P. These data satisfy one of the following conditions (for j - i and g - i ):
(03B1j) Cj has a one dimensional even theta characteristic L,. In this case the one
dimensional even theta characteristics on C are determined by their aspects
(see [E, H]1) ILil + (g - j)P on C, and ILg-, + 2PI + ( j - 2)P (where
Lg-j is any even theta-characteristic)
on
Cg- ¡.
17
The point P is in the support of an effective theta-characteristic Li on Cj.
Then, the aspects of the theta-characteristics on C are ILj + PI + (g - j - 1)P
and |Lg-j + 2PI + ( j - 2)P where Lg-i is any odd theta characteristic on
03B2j)
Cg - i.
Note: 03B11, 03B12,
Pl
are
empty.
(1.3). REMARK. The rational map from any component of 03B2j to Ai is genericany surjective. This follows from the fact that the locus in Ag of curves with
an
odd theta-characteristic of dimension greater than 0 has codimension 3
in
Ag (cf. [T]).
Proof of (1.2) : By the theory of limit linear series ([E, H]1), the curve C
will be in u1g if and only if there is a limit linear series of degree g - 1
and dimension one on C such that the corresponding line bundles on the
components C;, Cg-i of C differ from an even theta-characteristic on C only
up to tensoring with a divisor supported on the intersection point P. Write
(aj, bj) for the vanishing orders at P of the aspect of the limit linear series
on q. Then aj + bg-j
g - 1. One has 2bjP + 2Dj e |KCj(2(g - j)PI for
2. Hence, 2Di E |KCj(2cjP)| for
an effective D; and h0(Dj + (b; - aj)P)
c;
=
g - j - bj. Therefore,
Moreover, if equality between the right hand side and the left hand side
holds, b, = aj + 1 for j 1 for j = i and g-i.
Assume cj 0, then 2(D + P) E |KCJ|for an effective D. So, P is in the
support of an effective theta-characteristic on Cj. If Cj does not have a
=
one-dimensional theta-characteristic, then P is in the support of one of the
effective odd theta characteristics on Cj. Hence, fij is satisfied.
The locus of curves with a theta-characteristic of dimension at least 2 has
codimension 3 in Ap for every p (cf. [T]) and we are interested in loci of
codimension one in 0394i. Hence, if P is generic in Cj, Cj is in -4$’ as asserted
in oej.
Moreover, as the parity of a theta characteristic on a reducible curve is the
product of the parities of the theta-characteristics on its components ([F],
p. 14), it follows that the theta-characteristics on C are of the form stated
in (1.2).
Assume next ci
0. Then, Q are effective theta-characteristics
cg-j
and h0(D, + P) ~ 2. Hence, by Riemann-Roch, h0(Dj - P) h 1 and this
implies again that ç satisfies one of the conditions of (1.2).
Using the same approach, one finds that a generic curve with three
components is not in -9’ . The same will follow for a reducible curve with
=
=
18
two
p
components one of which is singular, once we know that, for
g - 1, the divisor Do of the moduli space of curves of genus p is not
contained in u1p. Assume this were not the case. Then, in particular, a curve
obtained by identifying a generic point of a generic curve of genus p - 1
with a point of a rational curve with a node would be in u1p. But the analysis
we have just done shows that this is impossible.
§2. Irreducibility
of
u1g
going to prove the irreducibility of JI; by induction on
being easy for g 5.
We
are
g, the result
(2.1 ). REMARK. In a neighborhood of a given point, the irreducibility of u1g
is equivalent to the irreducibility of the scheme which parametrizes curves
and even theta-characteristics of dimension at least one (see [T]). This
follows from the fact that this scheme has dimension 3g - 4 at all of its
points ([H], Th.1.10), all fibres of the map to u1g are finite and a generic
point in any component of JI; has only one theta characteristic of dimension one. ([T], Th.2.16).
(2.2). REMARK. On an hyperelliptic non-singular curve of genus g with
Weierstrass points R,, R2,
R2g+2’the theta-characteristics have the form
...
The dimension of this theta-characteristic is r. This expression is unique if
r 0, while in case r = - 1 the two expressions obtained by taking complementary sets of indices for the Weierstrass points give rise to the same line
bundle (use Riemann-Hurwitz Formula for the double covering of P1).
(2.3).
REMARK.
Every divisor in
Jtg intersects At.
enough to show that the generators of the Picard group of Jtg
independent elements in 0, .
Denote by kkl the moduli space of 1-pointed stable curves of genus k. From
[A, C] Prop. 1, the Picard group of the moduli functor for Jtk (k 3) has as
basis the classes ,1/, 03C8’, 03B4’0, 03B4’1, ... ,03B4’g. Here ,1’
039Bg03C0*(03C903C0), 03C8’ = 03C3*(03C903C0)
where 03C0:X ~ S, 03C3 : S ~ X is a family of 1-pointed stable curves. Also, 03B4’0
represents the boundary class of curves with a non disconnecting node and
Pro of.
It is
map to
=
19
b; the class of curves with a node which disconnects them in a piece of genus
i containing the section and a piece of genus g - i.
There is a clutching morphism from -1 x ..Ji" onto At. Consider the
composite map h
We consider the pull-back by h of the classes 03BB, 03B40, 03B41, 03B42 , ... , 03B4g/2 (or
03B4(g-1)/2). These may be expressed as tensor products of the classes in the
Picard functors
and vit.. with non-zero coefficients in 03BB’, 03B4’0, 03C8’ and
b’g - 2, 03B4’1 and 03B4g-3,
03B4’g/2-1 (or 03B4’(g - 3)/2 and 03B4’(g-1)/2) where all classes
refer to classes in
(cf. [A, C] Lemma 1 and [K] Th.4.2.).
of u1-1
...
,
u1g-1
(2.4). THEOREM: The divisor u1g is irreducible.
Proof: From (1.2), the intersection of -kgl and 03941 consists of 2 pieces 03B1g-1 =
a
03B2. By induction hypothesis, oc may be assumed to be irreducible.
Moreover, an infinitessimal calculation (see [T]), shows that only one
sheet of the scheme parametrising pairs of a curve and a one-dimensional
theta-characteristic contains a point (C, L) where C is a generic point of any
and 03B2g-1
=
of these components and C is a one-dimensional theta-characteristic on it.
Consider the curve C obtained by identifying a Weierstrass point P in a
generic hyperelliptic curve C’ of genus g - 1 with a point in a generic
elliptic curve E. This curve belongs to a. From (1.3), (2.2) and the fact that
the monodromy on the set of hyperelliptic curves interchanges. Weierstrass
points, it also belongs to any of the components of 03B2.
Denote by R, , R2, R3 the point of E which differ from P by 2-torsion.
Denote by R4,
R2g+2 the Weierstrass points of C’ différent from P.
These are the ramification points of the limit g2 on C and hence are the limits
on C of the Weierstrass points on non-singular hyperelliptic curves.
The limit theta characteristics of type a on C are those whose aspect
on E differ from an even theta-characteristic on E in (g - 1)P, namely
those whose aspect on E is (9E(Pi + (g - 2)P) = (9E(Pj + Pk + (g - 3)P),
{i, j, k} = {1, 2, 3}. The limit theta-characteristics of type 03B2 are those
whose aspect on E corresponds to the line bundle (9E«g - 1)P
(9E(PI +
...
,
=
P2 + P3 + (g - 4)P).
The limit of a family of even effective theta-characteristics on nonsingular hyperelliptic curves is a theta-characteristic of type a or fi on C
depending only on how many of the Weierstrass points which are fixed
points in the moving theta-characteristic (see (2.2)) have as limits points on E.
From (2.2) and the fact that the monodromy on the set of hyperelliptic
curves
acts
transitively
on
the set of Weierstrass
points,
it follows that it
20
transitively on the set of theta-characteristics of a fixed given dimension
non-singular hyperelliptic curve.
Hence, the monodromy on the set of hyperelliptic curves interchanges any
theta-characteristic of type 03B2 with one of type a, except may be for the limit
of ((g - 1)/2)gz (when g - 3 mod. 4). By induction hypothesis, the monodromy on a acts transitively on the even effective limit theta-characteristics
on C. Hence the proof will be complete when we show that the monodromy
on Jti brings the theta-characteristic ((g - 1)/2)gz to one of lower dimension.
To this end degenerate C to the curve
acts
on a
Here E’ is elliptic, P and P’ differ by 2-torsion on E’, C" is hyperelliptic and
Q is a Weierstrass point on it which has been identified with the point P’
on E.
Consider the family of curves Cx obtained from the join of the curves E
and E’ at the point P and the curve C" by identifying a variable point X in
E’ with the fixed point Q in E". Denote by P4, P5 the two points of E’ which
differ by 2-torsion from P and P’. Consider the theta-characteristic on CX
with aspects (g - 1)P on E, Q + (g - 2)X on E’ and (g - 1)Q on C". For
any X, there is a limit linear series of dimension at least one on Cx whose
Q, this is the limit of the
aspects correspond to these line bundles. For X
theta characteristic ((g - 1)/2)g§ on nearby hyperelliptic curves. For
X
R5, Cx is also hyperelliptic and the above aspects are the limit of a
theta-characteristic of the form R, + R2 + R3 + R4 + ((g - 5)/2)g2 on
nearby curves. This completes the proof of Theorem (2.4).
=
=
§3. Computation of the cohomology
Denote
by 03BB
and
ô, for
i
(3.1 ). PROPOSITION. The
=
0
class
...
class of
u1g in ug
[g/2] the basic divisor clases
of u1g
is
in Pic
(ug).
21
Write a03BB - 03A3[g/2]i=0 bi03B4i for the class of u1g.
Take two generic curves C,, C2 of genus i and g - i respectively,
i
g - i. Identify a fixed generic point P in C, with a moving point Q in
One
obtains in this way a curve F in Ai c JHK isomorphic to C2. This
C2.
0 for j ~ i, FJi = 2 - 2(g - i ) ([H, M] p. 86). On
satisfies F03BB
0, F03B4j
other
the
hand, by (1.2), the points of F in -Ù§ are obtained when Q is in the
support of an effective theta-characteristic on C,. For every such curve and
every effective theta characteristic on C,, one obtains a one dimensional
theta-characteristic on the reducible curve. As a generic curve of genus p has
2(p-I)(2P - 1) effective theta characteristics, one finds Fu1g = (g - i 1)
2(g-i-1)(2(g-i) - 1)2(i-I)(2i - 1). This gives the expression for b;, i &#x3E; 0.
The value of a and bo can now be obtained from the knowledge of bl and
b2 by using Theorem (2.1) in [E, H]2 and (1.2) above.
Pro of.
=
=
-
REMARK (of the referee): the coefficient of À, 2g-3(2g + 1), is what one
should expect from classical arguments. In fact, the product of the even
theta-characteristics is a modular form of weight 2g-2(2g + 1) (see [Fr]).
Moreover, Jtg is tangent to the pull-back of the divisor in Ag of abelian
varieties having a theta-null. The latter statement may be checked using the
interpretation of the tangent spaces to Ag and -4Yg at a point corresponding
to the (jacobian of the) curve C as the duals of S2(H0(C, KC)) and
H°(C, 2KC) respectively (see [0, S]).
Acknowledgements
1 would like to thank Gerald Welters and Joe Harris for some helpful
conversations during the preparation of this work and also Brown University for its hospitality.
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